A cumulative assignment scheme (CAS for short) is a special type of secret sharing schemes. For any given access structure (AS), a CAS which minimizes the cardinality of the primitive share set (the average information rate, or the worst information rate) is called an optimal CAS and can be constructed via solving some binary integer programming (BIP). The problem of finding optimal CAS\'s for complete AS\'s is solved.

We consider in this paper the problem of finding optimal CAS\'s for incomplete AS\'s. The paper introduces some notions including the connected-super-forbidden-family and the lower-forbidden-family for AS\'s. We show that an optimal CAS can be derived from some smaller sized BIP whose variables (constraints, resp.) are based on the connected-super-forbidden-family (lower-forbidden-family, resp.) of the given AS. The paper further builds the close relationship between the problem of finding optimal CAS\'s and the set covering problem (SCP). We prove that the problem of finding a CAS with minimum cardinality of the primitive share set (or minimum average information rate) is equivalent to the SCP, and thus is NP-hard. Other contributions of the paper include: 1) two types of AS\'s are recognized so that we can construct the corresponding optimal CAS\'s directly; and 2) a greedy algorithm is proposed to find CAS\'s with smaller worst information rate.

A $(t,n)$-threshold secret sharing scheme is a method to distribute a secret among $n$ participants in such a way that any $t$ participants can recover the secret, but no $t-1$ participants can.

In this paper, we propose two secret sharing schemes using non-abelian groups. One scheme is the special case where all the participants must get together to recover the secret. The other one is a $(t,n)$-threshold scheme that is a combination of Shamir\'s scheme and the group-theoretic scheme proposed in this paper.

In implementation of elliptic curve cryptography, three kinds of finite fields have been widely studied, i.e. prime field, binary field and optimal extension field. In pairing-based cryptography, however, pairing-friendly curves are usually chosen among ordinary curves over prime fields and supersingular curves over extension fields with small characteristics.

In this paper, we study pairings on elliptic curves over extension fields from the point of view of accelerating the Miller\'s algorithm to present further advantage of pairing-friendly curves over extension fields, not relying on the much faster field arithmetic. We propose new pairings on elliptic curves over extension fields can make better use of the multi-pairing technique for the efficient implementation. By using some implementation skills, our new pairings could be implemented much more efficiently than the optimal ate pairing and the optimal twisted ate pairing on elliptic curves over extension fields. At last, we use the similar method to give more efficient pairings on Estibals\'s supersingular curves over composite extension fields in parallel implementation.

Bitcoin is a decentralized payment system that is based on Proof-of-Work. Bitcoin is currently gaining popularity as a digital currency; several businesses are starting to accept Bitcoin transactions. An example case of the growing use of Bitcoin was recently reported in the media; here, Bitcoins were used as a form of fast payment in a local fast-food restaurant.

In this paper, we analyze the security of using Bitcoin for fast payments, where the time between the exchange of currency and goods is short (i.e., in the order of few seconds). We focus on double-

spending attacks on fast payments and demonstrate that these attacks can be mounted at low cost on currently deployed versions of Bitcoin. We further show that the measures recommended by Bitcoin developers for the use of Bitcoin in fast transactions are not always effective in resisting double-spending; we show that if those recommendations are integrated in future Bitcoin implementations, double-spending

attacks on Bitcoin will still be possible. Finally, we leverage on our findings and propose a lightweight countermeasure that enables the detection of double-spending attacks in fast transactions.

The Tardos code is a much studied collusion-resistant fingerprinting code, with the special property that it has asymptotically optimal

length $m\\propto c_0^2$, where $c_0$ is the number of colluders.

In this paper we simplify the security proofs for this code,

making use of the Bernstein inequality and Bennett inequality instead of

the typically used Markov inequality. This simplified proof technique also slightly improves the tightness of the bound on the false negative error probability. We present new results on code length optimization, for both small and asymptotically large coalition sizes.

Identity based encryption (IBE) has received great attention since Boneh and Franklin\'s breakthrough work on bilinear group based IBE [4]. Till now, many IBE schemes relying on bilinear groups with dierent properties have been proposed [5, 25, 29, 14]. However, one part of the user\'s private key in all these IBE schemes is constructed as y = f(msk), where msk is the master key and y is an element in the underlying bilinear group G. In this paper, we propose a new IBE: one part of the private key is y = f(msk), where msk is the master key and y is an element in Zp . Here p is the underlying bilinear group\'s prime order. By using some novel techniques, we prove this new IBE is semantic secure under the selective identity chosen plaintext attacks (IND-sID-CPA) in the standard model. Based on this IBE scheme, we construct

Secret sharing schemes usually have perfect information theoretic security. This implies that each share must be as large as the secret. In this work we propose a scheme where the shares are smaller, while the security becomes computational. The computational security assumption is hardness of factorization, which is a simple and rather standard assumption in cryptography. In our scheme, the shared secret can only be a set of prime numbers.

The Rainbow Signature Scheme is a non-trivial generalization of the well known Unbalanced Oil and Vinegar (UOV) signature scheme (Eurocrypt \'99) minimizing the length of the signatures. By now the Rainbow Band Separation attack is the best key recovery attack known. For some sets of parameters it is even faster than a direct attack on the public key. Unfortunately the available description of the attack is very ad hoc and does not provide deep insights.

In this article we provide another view on the Rainbow Band Separation attack using the theory of equivalent keys and a new generalization called good keys. Thereby we generalize the attack into a framework that also includes Reconciliation attacks. We further formally prove the correctness of the attack and show that it does not only perform well on Rainbow, but on all multivariate quadratic (MQ) schemes that suffer from missing cross-terms. We apply our attack and break the Enhanced STS signature scheme and all its variants, as well as the MFE encryption scheme and its variant based on Diophantine equations. In the case of Rainbow and Enhanced TTS we show that parameters have to be chosen carefully and that the remaining efficiency gain over UOV is small. As there is still some room to improve the Band Separation attack, it is not clear whether layer-based MQ-schemes will eventually become superfluous or not.